Prove the following statements.1. If a set of vectors contains exactly 2 vectors and those vectors are scalar multiples of each other, thenthe set is linearly dependent.2. If a set of vectors contains at least 2 vectors that are scalar multiples of each other, then the set islinearly dependent.3. If a set contains p vectors in R" and p> n, then the set is linearly dependent.4. If a set of vectors contains the zero vector, then the set is linearly dependent.5. If a set contains exactly one vector and it is nonzero, then the set is linearly independent.6. A set of vectors is linearly independent if and only if at least one vector in the set can be written as alinear combination of other vectors in the set.

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. Determine if these vectors are linearly independent or linearly dependent. If they are linearly independent show this. If they are linearly dependent, write a nontrivial linear combination (at least some coefficients need to be nonzero) of them being equal to 0. {4.0-6)}
The set of vectors {(1,1,0,0),(0,-1,0,0),(0,0,1,1),(1,0,1,1)} is linearly dependent. What is a valid expression of the last vector as a linear combination of the others? a. (1,0,1,1)=(1,1,0,0)+(0,-1,0,0)-(0,0,1,1) b. (1,0,1,1)=(1,1,0,0)+(0,-1,0,0)+(0,0,1,1) c. (1,0,1,1)=-(1,1,0,0)+(0,-1,0,0)+(0,0,1,1) d. (1,0,1,1)=2(1,1,0,0)+2(0,-1,0,0)+2(0,0,1,1)
2. If a set of vectors contains at least 2 vectors that are scalar multiples of each other, then the set is linearly dependent.
Question 5. Suppose that f is a Möbius transformation which fixes the real and imag- inary axes, i.e. f({ z = C: Im(z) = 0 }) = { z = C: Im(z)=0} and ƒ({ z € C: Re(z) = 0 }) = { z = C: Re(z) = 0}. Prove that f maps circles centred at the origin to circles centred at the origin (not necessarily of the same radius), i.e. for all r₁ > 0, there exists r2 > 0 such that ƒ({ z ≤ C : |z| = r₁ }) = { z € C : |Z| = r₂}. A geometric argument is sufficient, as long as you clearly state any properties of Möbius transformations that you use. You may assume, should you wish, that a circle is centred on a line if and only if it intersects the line at right angles.
4. ) Determine whether the following three vectors are linearly independent or inearly dependent. v₁ = (3,3, 0, 2), v₂ = (0, 2, 3, 1), V3 (3, 0, -2, -1). Hint: You can turn them into column vectors. =
Let u1, U2, ..., un be a set of vectors, and suppose there are nontrivial solutions to x11x2u2+ +xnun = 0 Explain why, if x is a free variable, you can always express u; in terms of the vectors corresponding to the basic variables.
There are 6 linearly independent vectors in R5. O True False
Show that a set of vectors, which contains a set of linearly dependent vectors, is linearly dependent. What is the analogous statement about linearly independent vectors?
Prove that two vectors are linearly dependent (taken as a set of vectors) if and only if one vector is multiple of the other.

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1. If a set of vectors contains exactly 2 vectors and those vectors are scalar multiples of each other, then the set is linearly dependent.